Weighted Envy Freeness With Bounded Subsidies
Noga Klein Elmalem, Rica Gonen, Erel Segal-Halevi
TL;DR
The paper introduces weighted-envy-freeness (WEF) for allocating indivisible items among agents with heterogeneous entitlements and shows subsidies can enforce WEF when given allocations are not inherently fair. It characterizes WEF-ability through the weighted envy graph $G_{A,w}$ by the absence of positive-cost cycles and develops polynomial-time algorithms to produce WEF-able allocations with provable subsidy bounds in three valuation regimes: general additive, identical additive, and binary additive. The key results include tight subsidy bounds for the general case with a given allocation, a weight-independent bound for integer weights with a constructive algorithm, and specialized bounds for the identical and binary valuation settings ($(n-1)V$ and $\frac{W}{w_1}-1$, respectively). These findings extend prior unweighted subsidy results to the weighted setting and provide practically efficient methods for ensuring fairness when entitlements differ across agents. The work lays groundwork for further exploration of subsidies under more general valuation classes and nonadditive models.
Abstract
We explore solutions for fairly allocating indivisible items among agents assigned weights representing their entitlements. Our fairness goal is weighted-envy-freeness (WEF), where each agent deems their allocated portion relative to their entitlement at least as favorable as any other's relative to their own. In many cases, achieving WEF necessitates monetary transfers, which can be modeled as third-party subsidies. The goal is to attain WEF with bounded subsidies. Previous work in the unweighted setting of subsidies relied on basic characterizations of EF that fail in the weighted settings. This makes our new setting challenging and theoretically intriguing. We present polynomial-time algorithms that compute WEF-able allocations with an upper bound on the subsidy per agent in three distinct additive valuation scenarios: (1) general, (2) identical, and (3) binary. When all weights are equal, our bounds reduce to the bounds derived in the literature for the unweighted setting.